On generalized Peano derivatives

Abstract

A function F F is said to have a generalized n n th Peano derivative at x x if F F is continuous in a neighborhood of x x and if there exists a positive integer k k such that a k k th primitive of F F in the neighborhood has the ( k + n ) (k + n) th Peano derivative at x x ; and in this case this ( k + n ) (k + n) th Peano derivative at x x is proved to be independent of the integer k k and the k k th primitives, and is called the generalized n n th Peano derivative of F F at x x which is denoted as F [ n ] ( x ) {F_{[n]}}(x) . If F [ n ] ( x ) {F_{[n]}}(x) exists and is finite for all x x in an interval, then it is shown that F [ n ] {F_{[n]}} shares many interesting properties that are known for the ordinary Peano derivatives. Using the generalized Peano derivatives, a notion called absolute generalized Peano derivative is studied. It is proved that on a compact interval, the absolute generalized Peano derivatives are just the generalized Peano derivatives. In particular, Laczkovich’s absolute (ordinary) Peano derivatives are generalized Peano derivatives.